Another way to do this, perhaps slightly simpler, is to subtract the equations.

You have x2 + x3 = b1, x1 + x3 = b2, and x1 + x2 = b3. Subtracting the first equation from the second eliminates x3: x1- x2= b1- b2. The third equation does not have x3 and just adding it to this new equation eliminates x2: 2x1= b1-b2+b3. x1= (1/2)(b1-b2+b3). You can now put that value of x1 back into the third equation and solve for x2. Once you have both x1 and x2, putting those expression into either of the first two equations gives and equation for x3.